ME 543 (CFD)
Homework Assignment 1, Marks 7
Submit by 2.00 pm, 7 September 2012
Note:
(i) You are required to enclose a printout of the code(s) (FORTRAN or C) you have
developed.
(ii) Please write down the date of submission conspicuously on the cover pag
Rectangular staggered finite-volume grid
and pressure-correction methods
FVM for 2D diffusion problem
n
W
w
P
e
y
E
s
S
x
2D steady state diffusion equation:
S 0
x x y y
(1)
Integrating the equation over the control volume we obtain
x x dxdy y y
ME 543 (CFD)
Home work assignment 3
Marks: 6
Submit by 19/10/2012
Note:
(i) You are required to enclose a printout of the code(s) (FORTRAN or C) you have developed.
(ii) Please write down the date of submission conspicuously on the cover page of your repo
Stability analysis of transient solutions of some parabolic PDEs
An FDE may be consistent but its solution will not necessarily converge to the solution of the PDE. The
Lax equivalence theorem states that a stable numerical method must also be used.
Consi
ME 543 (CFD)
Home work assignment 2
Marks: 6
Submit by 28/9/2012
Note:
(i) You are required to enclose a printout of the code(s) (FORTRAN or C) you have developed.
(ii) Please write down the date of submission conspicuously on the cover page of your repor
The first-order wave equation in one dimension
Let us now direct our attention to the first-order wave equation
u
u
c
0,
t
x
c0
The exact solution to this equation is
u t n t , x u t n , x c t
For the initial data u ( x, 0) F ( x),
x , the solution can
ME 543 (CFD)
Home work assignment 2
Marks: 6
Submit by 28/9/2012
Note:
(i) You are required to enclose a printout of the code(s) (FORTRAN or C) you have developed.
(ii) Please write down the date of submission conspicuously on the cover page of your repor
Stability analysis of transient solutions of some parabolic PDEs
An FDE may be consistent but its solution will not necessarily converge to the solution of the PDE. The
Lax equivalence theorem states that a stable numerical method must also be used.
Consi
Governing Equations of Fluid Dynamics
Anoop K. Dass
Dept of Mechanical Engineering
Indian Institute of Technology Guwahati
Guwahati, ASSAM
1
System of coordinates
Coordinates
x1
x2
x3
x
y
z
Unit vectors
i1
i2
i3
i
j
k
Velocity components
u1
u2
u3
u
v
w
Ve
Finite Differences
Let us consider the PDE 2u
2u 2u
0
x 2 x 2
(The Laplace equation)
If we intend to use a computer to solve this equation, we have to first discretize it on what is
known as a grid (See the figure below for a finite difference grid).
x
Solution Algorithms
Discretization of PDEs in the computational domain results in a system of algebraic
equations that needs to be solved numerically. The solution methods for the system of
linear equations can be broadly classified into two types direct
Mathematical classification of PDEs
The classification of PDEs is based on the mathematical concept of characteristics that are lines (in 2D) or
surfaces (in 3D) along which certain properties remain constant or certain derivatives may be
discontinuous. I
ME 543 (CFD)
Homework Assignment 1, Marks 10
Submit by 2.00 pm, 7 September 2012
Note:
(i) You are required to enclose a printout of the code(s) (FORTRAN or C) you have
developed.
(ii) Please write down the date of submission conspicuously on the cover pa
FD Equations or Difference Representation of PDEs
The finite difference approximations discussed earlier are used to replace the derivatives that appear in
the PDEs. Consider an example involving time t and a space coordinate x, to be precise, the heat eq
Classification of PDEs
Many important physical processes in nature are governed by partial differential equations
(PDEs). It is important to know the physical significance and mathematical behaviour of the
PDEs encountered in fluid mechanics and heat tran